论文标题
非Busemann Horoballs的示例
Examples of Non-Busemann Horoballs
论文作者
论文摘要
在F.G.设置，我们构建了horoballs，它们“不围绕着大地的海森堡组和所有带有无限表演组的花圈产品。也就是说，我们发现的是球的限制 - 称为Horoballs，这并没有增加地球上的点周围的球，称为Busemann Horoballs。就花圈产品而言，这是由于存在脱节的霍巴尔（Horoballs）的更强事实。在Lamplighter组上，我们展示了Busemann Horoballs的限制，这些Horoballs甚至没有连接。对于普遍的海森堡群体，我们改用对称论点。实际上，至少在特定的生成套件下，对于$ 3 $维的Heisenberg Group，我们表明所有Horoballs都是连接的。 该说明是与文献隔离的一定程度上写的，以回应最近的Epperlein和Meyerovitch的预印本，要求提供非Busemann Horoballs的例子。事实证明，上述大多数结果都是已知的：有关连接性的结果，从有关凸度的已知结果遵循。 Lamplighter集团中的非共同连接的Horoballs，以及Heisenberg Group中连接的非Busemann Horoballs，可能是新的陈述，但我们在这里也指出了一些密切相关的文献。
In the f.g. setting, we construct horoballs which are "not centered around a geodesic" for generalized Heisenberg groups and all wreath products with an infinite acting group. That is, we find are limits of balls -- called horoballs -- which are not increasing unions of balls around points on a geodesic -- called Busemann horoballs. In the case of wreath products this follows from the stronger fact that there exist disconnected horoballs; on the lamplighter group we exhibit limits of Busemann horoballs which are not even coarsely connected. In the case of generalized Heisenberg groups, we use a symmetry argument instead; in fact for the $3$-dimensional Heisenberg group, at least under a particular generating set, we show that all horoballs are connected. This note was written somewhat in isolation from the literature, in response to a recent preprint of Epperlein and Meyerovitch asking for examples of non-Busemann horoballs. It turns out that most of the results mentioned above are known: The results about connectedness follow from known results about almost convexity. The non-coarsely connected horoballs in the lamplighter group, and connected non-Busemann horoballs in the Heisenberg group, may be new statements, but here also we point out some strongly related literature.
